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Harmonic Segments Along Fifths (HSAF) [sing. ''a Harmonic Segment along Fifths''] are a type of octave-[[periodic scale]] built by taking a specific [[harmonic series segment]] that ends in 4/3 and repeating times 3/2. They are a unique mix of [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).
Harmonic Segments Along Fifths (HSAF) [sing. ''a Harmonic Segment along Fifths''] are a type of octave-[[periodic scale]] built by taking a specific [[harmonic series segment]] that ends in 4/3 and repeating times 3/2. They are a unique mix of [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation).
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* Its prime mode always contains 4/3 and 3/2.
* Its prime mode always contains 4/3 and 3/2.
* Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment is repeated transposed by 3/2 until it reaches the octave.
* Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment is repeated transposed by 3/2 until it reaches the octave.
* Its [[delta signature]] is +2,+2,+2,+2,[…],+3,+3,+3,[…]. If the HSAF is even-indexed, the +3's start right at 3/2. If the HSAF is odd-indexed, at the step right before 3/2.
* HSAF 1 is 6:8:9:12 with [[delta signature]] +2,+1,+2. Except for HSAF 1, its delta signature is +2,+2,+2,+2,[…],+3,+3,+3,[…]. If the HSAF is even-indexed, the +3's start right at 3/2. If the HSAF is odd-indexed, at the step right before 3/2.


These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSAF are a countable set of scales, and thus can be indexed.
These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSAFs are a countable set of scales, and thus can be indexed.


== Properties ==
== Properties ==
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=== HSAF 2 - ''Zontatonic'' ===
=== HSAF 2 - Septimal minor pentatonic ===
<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.  
<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.  


First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
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=== HSAF 3 - ''Íegmul'' ===
=== HSAF 3 ===
<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.  
<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.  


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=== HSAF 4 - ''Ngwóoghe'' ===
=== HSAF 4 ===
<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.
<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.


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=== HSAF 8 - ''Veljeuwof'' ===
=== HSAF 8 ===
<code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>. Harmonic segment: '''24::32.'''
<code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>. Harmonic segment: '''24::32.'''


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